Position measuring through the use of moire fringe multiplication

ABSTRACT

A system for linear and angular measurements utilizing a form of moire fringe multiplication particularly in position measuring apparatus. The position of a table is determined by moire fringe multiplication with a coarse scale grating, an index member in juxtaposition with said scale grating, illumination, optical filtering, and observing means and means to indicate relative position of the table from changes of the said moire fringes.

United States Patent 1191 Mar. 12, 1974 Post i POSITION MEASURINGTHROUGH THE USE OF MOIRE FRINGE MULTIPLICA'IION [76] Inventor: DanielPost, Box 408, Averill Park,

[22] Filed: May 14, 1971 211 App]. No.2 143,561

[30] Foreign Application Priority Data Oct. 19, 1970 Great Britain49,478/70 52 US. Cl. 356/169, 250/237 0 [5]] Int. Cl. .Q G0lb 11/04 [58]Field ofSearch 356/106 R, l69, 151;

[56] References Cited UNITED STATES PATENTS 2,886,718 5/1959 Shepherd etal .1 356/169 3,482,107 12/1969 Hock 356/169 3,572,937 3/1971 Baldwin356/110 3,574,292 4/1971 Butts 356/106 3,586,665 6/[971 Weyrauch...356/169 3,588,462 6/1971 Kreckl 356/l69 OTH ER PUBLICATIONS Jenkins andWhite, Fundamentals of Optics, 3rd Ed.,

McGraw-Hill Book Company, Inc, NY. (1957) pp. 239-441.

Langenbeck, Higher-Order Lloyd interferometer, Applied Optics, Vol. 9,No. 8 (August 1970) pp. 1838-1841.

Oster, Moire Processing of Biological Data, Annals of the NY. Academy ofSciences, Vol. 157, Art. 1 (March 31, 1969) pp. 83-96.

Primary Examiner-William L. Sikes Attorney, Agent, or Firm-Karl W.Flocks 57] ABSTRACT A system for linear and angular measurementsutilizing a form of moire fringe multiplication particularly in positionmeasuring apparatus. The position of a 16 Claims, 8 Drawing FiguresSIGNAL PULSE 1 TABLE TABLE 006113120111- A' MOTION POSITIONING POSITIONoevme INDICATOR CONTROL PAIENIED R 2 8 3. 796.498

SHEU 1 BF 3 I4 I I5 22 M23 5% T I 24 I H I I3 I i/.25

\l -F/GI 2. 28 I 26 29 I SIGNAL PULSE TABLE TABLE COI\IIIRJIEON-.,COUB"ITER MOHON I POSITIONING POSITION DEVICE CIRCUIT 'INDICATORCONTROL 3Q -3 l 32 -33 INVENTOR DANIEL POST ATTORNEY PAIENIEDIAR t 21974 3L 7 96; 498

SHEET 2 BF 3 FIG. 3. 35 24 23 22 l i ll l 36XTQE L 25 27 26 FIG. 4.

23 22. was X' [\,27 .Llw 29 FIG. 5.

GRATINGS T INDEX SCALE 3 (I I) up) OBSERVING SYSTEM INVENTOR DAN I E L P081' ATTORNEY PAIENIEDMARIZIBM V 3,795,498

SM] 3 0F 3 FIG. 6. W

T 31 FIRST g ORDER /73 ZEROTH I ORDER a 72 7| b SNELL "'REFRA0T|0 Fla 2'DIRECTION an X 82 g/2) 83 (0) 3 l (o, p/2,o) (0454 FIG 8. r

INVENTOR DANIEL POST BY QQ- KaQv ATTORNEY POSITION MEASURING THROUGH THEUSE OF MOIRE FRINGE MULTIPLICATION BACKGROUND OF THE INVENTION In theprior art, precision linear positioning and mea suring has beenaccomplished by the following methods; angular positioning and measuringhas been accomplished by variations of these methods:

1. Precision screw and nut devices. With this method, precision screwsmust each be manufactured and corrected as originals-no replicatingprocess is applicable and cost is high. Screws and nuts are susceptibleto changing accuracy with age from wear and changing frictioncharacteristics. Periodic maintenance principally cleaning andlubrication is maximal. The system is highly susceptible to errorassociated with variations of forces in machine tool operations. 2.Scale and index mark devices. This system is characteristically highlystable. Its primary source of error stems from random irregularities inpositions of markings on the scale. I

3. Moire scale and index devices. This method is similar to 2 above, butthe scale and index elements are diffraction gratings, with rulingfrequencies usually equal, but sometimes in the ratio of small integers,e.g., 1:2. The outputis in the form of mechanical or opticalinterference fringes, called moire fringes. Small relative motion of thegratings induces large motion of the fringes. These fringes are readilysensed by photodetectors and interpreted by pulse counters, andtherefore the moire method is highly compatible with automatedmeasurement and control systems. A fundamental advantage of moire is theaveraging effect, wherein the output signal depends upon the averageposition of all the index grating rulings relative to the adjacent scalegrating rulings. Random irregularitiesin positions of rulings arecancelled; they have no effect on moire fringe position, and only minoreffect on fringe contrast. A primary disadvantage of the moire method isthe low sensitivity of systems employing relatively inexpensive scalegratings, and conversely, the extremely high cost and fragile nature ofscale gratings required for systems of high sensitivity.

4. Optical interferometer devices. The advent of the laser has allowedoptical interferometers to become practical measuring devices forindustrial applications.

' Interferometers have the advantage of being highly sensitive, ca y.2199! is 'q-i shq P9! f n e- However, they are highly susceptible toerrors i ntfo duced by variations of wavelength of the light employed,variations of refractive index of the air in the space separatinginterferometer mirrors, and out-ofplane vibrations of interferometermirrors. The provisions required to minimize these sources of errorincluding stabilized lasers, barometric pressure compensation,environmental control are extremely expen sive. Still, these errorslimit the accuracy of laser interferometry to about one part permillion, whereas the theoretical accuracy is orders of magnitude greaterfor measurements of long paths. The system is restricted,

too, in that its basic range of sensitivity is not widely adjustable,but limited by the availability of suitable light sources.

SUMMARY OF THE INVENTION The object of the present invention is toprovide a highly versatile, highly sensitive and highly stable systemfor linear and angular measurements. The system utilizes special formsof moire fringe multiplication. Sensitivity can closely approach that ofoptical interferometry while the stability inherent in mechanical scalesis maintained. The system has special applicability to themicroelectronics industry, which requires ultra precise positioning andmeasuring: across moderate distances and the machine tool industry,which requires precise positioning and measuring across large distances.I

The moire fringe multiplication as performed in this invention providesone to two orders of magnitude increase of sensitivity of the moiremethod, while retaining a relatively coarse, robust and inexpensivescale grating. The sensitivity is readily variable, and it can closelyapproach the sensitivity of a double path optical interferometer.Accuracy exceeding one part per million is practical.

The measuring system is not subject to wear or degradation with age; itis insensitive to random errors of ruling positions; its sensitivity andaccuracy are not unduly influenced by wavelength and wavelength spreadof the light source, nor by refractive index and barometric pressure ofthe surrounding medium, nor by out-ofplane vibrations of the gratings.The scale grating can be made very long.The scale grating and indexmember can be separated by a gap sufficiently large for convenientset-up and operation. The moire fringes and output signal areindependent of changes of thickness and taper of the gap between scalegrating and index member; they depend only upon the in-plane componentof motion of the scale grating normal to its rulings.

BRIEF DESCRIPTION OF THE DRAWINGS FIG. 1 illustrates the externalappearance of one form of this apparatus;

FIG. 2 is a schematic representation of one form of the optical systemand signal processing components;

FIG. 3 is aschematic representation of an alternate arrangement of theoptical system;

FIG. 4 is a schematic representation of another alternate arrangement ofthe optical system;

FIG. 5 illustrates the diffraction of light for the optimumconfiguration of moire fringe: multiplication elements;

FIG. 6 is a cross-sectional view illustrating one type of index gratingthat produces neighboring doubleorder dominance;

FIG. 7 is a cross-sectional view illustrating another type of indexgrating that produces neighboring double-order dominance; 7

FIG. 8 illustrates the diffractions experienced by dominant. rays forthe reflection equivalent of the optimum configuration.

DESCRIPTION OF THE PREFERRED EMBODIMENTS In FIG. 1, table 11 translateswith respect to its base 12. A scale grating 13 is attached to saidtable with its width parallel to the axis of translation. Housing .14,which is fixed to base 12, contains an index grating with rulingsapproximately parallel to the rulings of the scale grating, (or itcontains an equivalent prism) and an illuminating, optical filtering andreceiving system. Cable provides conductors for power input and signaloutput.

Two such systems can be assembled together for biaxial motions and threefor triaxial motions. Similar systems may be used for angulardisplacements, wherein the scale grating is circular and concentric withthe center of rotation; the scale grating may be cylindrical, withrulings parallel to the axis of rotation, or it may be plane, withrulings radiating from the axis of rotation.

Contents of housing 14 are represented in FIG. 2. These comprise indexgrating 21 mounted essentially parallel to scale grating 13. Light fromconcentrated source 22 is rendered approximately monochromatic by filter23 and collimated by lens 24, so as to impinge on the index grating inthe desired direction of incidence. Emergent light is decollimated bylens 25 and focused to form an image of moire fringes on image plane 26.Aperture plate 27 is located in the focal plane of lens 25 and functionsas an optical filter by passing the desired family of diffracted raysand stopping all other families. Aperture 28 in image plane 26 is a slitorientated parallel to the moire fringes, allowing light into thephotodetector 29. As scale grating 13 translates with respect to indexgrating 21, the moire multiplication effect causes alternately intenseand dim light to fall on slit 28, and in turn, causes correspondinglyhigh and low electrical output from photodetector 29.

This modulated electrical output is treated by a signal conditioningcircuit 30 and fed into a pulse counter and position indicator 31, knownin the prior art. The modulated signal may also be fed into a cablemotion controller 32, and cause it to actuate a table positioning device33 according to a predetermined program.

The pulse counter may utilize two observations and signals from themoire pattern, for example, observations at locations one-fourth moirefringe apart. Then, two slits may be used in the image plane, and twophotodetectors may be used according to techniques known in the priorart. Other known techniques utilizing more than one optical signal maybe used to afford fringe interpolation and discrimination betweenforward and backward movement of the scale grating.

In FIG. 3, mirrors 35 and 36 are used to fold the optical path forapplications in which a more compact physical design is desired. Otheroptically equivalent but physically different designs are possible.

In FIG- 4 the apparatus is arranged to observe the moire fringes formedby light reflected back from the gratings.

A primary contribution of this invention is the optical configuration ofa particular moire fringe multiplication system, ideally suited forposition measuring apparatus. This optimum configuration is specified infollowing paragraphs and tests of its performance are describedthereafter,

OPTIMUM OPTICAL CONFIGURATION The optical configuration is illustratedin FIG. 5. The fine index grating 21 exhibits dominant diffraction intwo neighboring orders, usually the zeroth and first orders but forgenerality, let the dominant orders be a and a+l. Illumination is by abeam of collimated light whose plane of incidence is perpendicular tothe rulings. Angle of incidence a is so arranged that the normal to theindex grating bisects the angle between dominant diffraction orders.Then, angle of incidence is given by sin a -(a+ /z))\/g, (I)

and the dominant diffraction angles by sin .1 g1 (2) where A iswavelength of the monochromatic light employed and g, is pitch of theindex grating; acute angles are positive if rotation from the referenceaxis to the ray is counterclockwise; diffraction angles increase as aincreases. Since 0,, ='6,, (3)

light of the dominant diffraction orders is symmetrically disposed withrespect to the grating normal. In practice, the prescribed angle ofincidence usually can be set by adjusting the position of back-reflectedimages of the source with respect to the source itself.

The pitch of the coarse scale grating is approximately B times greaterthan the pitch of the reference grating, where ,B is a whole numberwhich is not necessarily small, for example, a number between 3 and 100;in the Figure B 6. ,8 is defined as the fringe multiplication factor.

The coarse or scale grating should preferably exhibit symmetricaldiffraction intensities with respect to its zeroth order. This conditionis automatically satisfied with an amplitude grating (Ronchi ruling orbar-andspace grating with opaque or semi-transparent bars) and with abar-and-spac-e phase grating (transparent bars typically producingone-half cycle phase retardation). The coarse or scale grating ispositioned on the observer side of the fine or index grating, andoriented with nearly parallel grating surfaces and nearly parallelrulings. The gap between gratings would usually be small, for example,from zero to one mm.

Parenthesized numerals attached to diffracted rays in FIG. 5 indicatethe diffraction orders experienced by the ray, and a pair of numbersseparated by a comma indicate diffraction orders experienced at theindex and scale gratings, respectively. I

Angles of diffraction, 'y, of light emerging from the scale grating,measured from the normal to the grating, are given by sin y qlt/g sin0,, 4

where p and q are'the corresponding diffraction orders experienced atthe index and scale gratings, respectively, and g. is pitch of the scalegrating. In practice, the pitches are related, either exactly or verynearly,by

Thus, rays of order sequence (a, B/2) and (a l,8/2) are found bycombining Eqs. 2, 4 and 5 to emerge in fhe directiony ii They emergepafallelto the axis of the optical system and converge at point 0 on theoptical axis of the decollimating lens 25. This is illustrated in FIG. 5for rays of order sequence (0,3) and 1,-3), since here a O and B 6.

Similarly, pairs of rays that experience orders (a, 8/2 +s) and (a l,B/2 s), where s is a whole number,

orders at thefine grating also become members of each s-group-forexample rays of l, 9 and 2, -9 diffraction orders in FIG. 1 enter the sgroup. However, these members carry negligible energy compared to thestrong members, and are disregarded. Aperture plate 27 is located in thefocal plane of lens 25, and functions as an optical filter so as toisolate light in any s-group for observation. In practice, this apertureplate is usually positioned to admit rays of groups s 0.

TESTS OF PERFORMANCE Dependence Upon S-Group The optical arrangementdepicted in FIGS. 2 and 5 was used. The index or fine grating 21 wasphase type with 1,200 rulings per mm, exhibiting dominant diffraction inzeroth and first orders. The coarse scale grating 13 was a Ronchi rulingof 20 rulings per mm. This barand-space ruling consisted of opaque barsof evaporated chromium on a glass substrate; widths of opaque bars andclear spaces were equal.

The index grating was 5 cm diameter and the specimen grating was 2.5 cmlong by cm wide. A field of view approximately 5 cm wide and 2.5 cm highwas observed.

Paper spacers were placed between the gratings, providing a wedge-shapedgap between the grating surfaces. Gap thickness was 0.05 mm on one endand 0.10 mm on the other end 5 cm away. The apex of the wedge-shaped gapwas approximately parallel to the rulings.

The scale grating was rotated in its plane to minimize the number ofmoire fringes across the field. Seven fringes appeared inthe field ofview when group s 0 was observed. Fringe contrast was high.

Observations were made in numerous s-groups surrounding zero. Thedecollimating lens 25 and aperture plate 27 were mounted on a commontrack that pivoted about a point near the gratings. Thus, the track'wasrotated to the position where the aperture plate admitted the desiredsgroup. An image of the moire pattern was formed in a. camera locatedbehind aperture plate 27 and observed with a focusing magnifier in thecamera image plane.

Fringes had sufficient contrast in s groups -10 through 13 formeaningful interpretation. No changes were observed in fringe count,inclination or shape, over the whole range of observations.

Fringe contrast varied with s-group. Also, the positions of intensityminirna varied with s-group observed-notably, positions of intensitymaxima and minima interchanged in odd and even s-groups near s 0. Sinceone is always interested in the difference between moire fringe ordersappearing in one particular 5- group, this shift of fringe centers isunimportant.

These results show that the moire pattern is independent of angularposition of observation near s 0 (when angle of incidence is asprescribed).

Sensitivity Separate observations were made with five different indexgratings 21 of frequencies 200 to 1,200 rulings per mm, mounted inseries with the rulings per mm chromium scale grating 13. Rays in groups 0 were isolated and observed. A micrometer drive was arranged forcontrolled rotation of the scale grating with respect to the fixed indexgrating. Moire fringes generated perpendicular to the rulings werecounted as a function of angular position.

Results showed that, at any point in the field,

. factor. From Eq. (2), the limiting value ofg, is M2. Ac-

cordingly, the maximum sensitivity obtainable is d M2 per moire fringe.The limiting sensitivity thus equals the sensitivity of opticalinterferometry with a double-path. interferometer. Dependence Upon GapEffects of finite gap between the grating surfaces were investigated.The gratings employed were the 1,200 rulings per mm index grating andthe 20 rulings per mm chromium scale grating, producing moiremultiplication by 60.

In one setof observations, the incident beam was adjusted to providesymmetrically diffracted zeroth and first order beams, as prescribed byEq. 2 and illustrated in FIG. 5 when 0 =0 In another set, the angle ofincidence, a, was altered to produce various ratios of 0 /6 In allcases, light that was isolated and observedwas the group whose dominantmembers experienced diffraction orders 0, 30 and l, 30 at the index andscale gratings, respectively. This was the condition of superior fringecontrast. n e

The gap and taper between grating surfaces were varied by introducingpaper spacers. Conditions of incidence and observation were set and thenthe Ronchi ruling was rotated in its. plane to minimize fringe densityacross the field. Thus, moirefringes from rotational mismatch wereeliminated and the residual fringes re sulted from slight extensionalmismatch (deviation from B and from gap effect, if any.

TABLE I below shows the experimental conditions and results. It can bededuced that seven fringes were caused by extensional mismatch alonewhen 0,, =0

. TABLE I Gap Effect Fringes Gap Gap Across Effect,

-6,,/0, mm Taper Field Fringes 1 0.05 0 7 t t 0 l 0.20 0 7 7 0 10.00-0.10 7 0 1 0.10-0.20 7 0 1 0.00-0.10 ,7 0 5/7 0.10 0 5 2 5% 0.10 t0 3.5 3.5

*indicates apex of wedge-shaped gap is perpendicular to rulings. andindicates apex is parallel to rulings. indicates no taper.

These results, taken together with the results of the section Dependenceon S-Group, show that number, inclination and shape of moire fringes areindependent of both gap and obliquity of observation, if 6 0,.Consequently, symmetry of the two dominant intergrating beams is theparamount condition for uniqueness of the contour map generated by moirefringes.

TABLE I also shows that gap and taper are responsible for extraneousfringes when 6., 6,. The dependence is moderate, however, when gap andtaper are small, and small deviations from the condition of symmetrywould have negligible influence.

ADDITIONAL PERFORMANCE CONSIDERATIONS Fringe Contrast The main causes ofdegradation of fringe contrast are:

l. unequal intensities of the two dominant diffractions from the indexgrating,

2. low signal-to-noise ratio in diffraction orders used to form thepattern,

3. excessive width of light source when gap between grating surfaces issignificantly large.

Unequal Intensities Ideally, the intensities of the two dominantdiffractions, a and a+l, should be equal, and they should be largecompared to the non-dominant diffractions. When mated with asymmetrically diffracting coarse grating, the beams that experiencediffractions a, ,B/2 and a+l, ,8/2 would have equal intensities andcombine in two-beam interference. Optimum contrast then appears when s0.

When intensities of a and a+l are unequal, contrast cannot be perfectfor s 0. However, considerable inequality can be tolerated beforecontrast becomes poor. For example, when two beams of intensity ratio4:1 enter into interference, the interference fringe intensity maximaand minima are in the ratio 9:1, which yields high contrast.

In addition, when contrast is not perfect with groups 0, it is likely tobe more nearly perfect in a nearby group. This occurs for diffractionsequences a, B/Z s and a+1, -B/2 s when the intensities, l, diffractedinto each order are most closely related by ll n+1 L B/2H-s -H B/2)+s Inpractice, the observer can survey the patterns for sgroups near s=0 andchoose the pattern of maximum contrast.

Signal-to-Noise Ratio When a grating is illuminated by collimatedmonochromatic light, most of the light emerges in a system of discretedirections. However, as a result of random errors existing throughoutthe ruling, a portion of the light scatters out in every direction. Thisscattered light can be considered as a background noise superimposedupon the signal of diffracted light. Similar effects are caused by dirtand blemishes and these add to noise in more localized regions.

This optical noise adds background intensity to the moire pattern. Incases where the intensity of the moire pattern itself becomes so feebleas to be overwhelmed by the background intensity, no fringes arevisible. In cases where signal and noise are of the same order ofmagnitude, fringes appear with reduced contrast; and where the fringesare very strong compared to the background intensity, the noise is of noconsequence.

The variation of intensity with diffraction order for pure bar-and-spacegratings is treated in elementary texts. Intensity diminishes rapidlyand is very small for high orders, e.g., for the twenty-fifth order.

Since diffraction orders of B/2 or slightly higher are used, the beamsthat form the moire pattern are relatively weak when B is a largenumber. Then, the noise level must be relatively low. Of the two, thecoarse grating is usually responsible for the greater contribution ofnoise. For large multiplication factor, i.e., large ,8, the coarsegrating must be clean and must have smooth, sharply defined bars andspaces. With the chromium Ronchi ruling used in the aforementionedexperiments, the signal-to-noise ratio was sufficiently high to producegood fringe contrast up to the eightieth order; with a photographicgrating of 20 rulings per mm, up to the twenty-fifth order; and withanother relatively crude photographic grating of 20 rulings per mm, onlythe tenth order.

The maximum multiplication factor with this chromium grating exceeds160. Excellent contrast was obtained in the aforementioned experimentsat B=60. With the said photographic gratings, maximum multiplicationfactors are about 50 and 20, respectively. Accordingly, the gratingsignal-to-noise ratio may be a limiting factor for multiplications neartwo orders of magnitude, while multiplications near one order ofmagnitude are virtually unrestricted.

When fringe contrast is significantly reduced by optical noise, someimprovement is afforded by minimizing aperture 27 in the focal plane ofthe decollimating lens. An aperture larger than that required forresolution of the moire fringes admits the scattered light (noise) in Ilarger proportion than the directed light (signal). Width of Source Fora given grating separation, the permissible source width decreases aspitch of the index grating decreases. When the said grating of 1,200rulings per mm was used with a gap of 0.05 mm, excellent contrast wasachieved with angular width of source of H700. Noticeable contrastdegradation occurred with twice this source width. An angular width ofsource of H500 was used with all the other aforesaid gratings, sincethis produced sufficient illumination for convenient observation.

In practical applications of moire fringe multiplication, there wouldseldom be need for a gap greater than 0.05 mm. Under this condition, thepermissible width of source is sufficiently large that contrastdegradation can be easily avoided.

Index Grating Design Blazed Phase Gratings A phase grating blazed fordominance in two neighboring orders may be used for the index grating.FIG. 6 shows the prescribed directions of incidence (d) and emergence ofzeroth and first order diffractions (0 0 for the fine reference grating.Diffraction angles depend upon grating pitch and wavelength ofillumination. Grating blaze angle, a, controls the energy distribution.If (I) is such that the natural direction of refraction-as calculated bySnells Law-is the same asthe sin (dr-l-a) n sin (dz-fa) a-0,, where n isrefractive index of the grating material and where clockwise angles inthe figure are negative.

Refraction angle 00' was calculated for the fivereference gratings usedin the aforesaid experiments, with illumination at wavelength A 5,461 A.The ratio of refraction to diffraction angles, 0,, (0 -0 is listed inTABLE 1] below.

When thickness h of the grating layer is related to width w betweenadjacent bars by w 2hR sina/ Wi -sin er 1 1) where n is the index ofrefraction of the transparent zone 74, equal intensities emerge in thezeroth and first diffraction orders. Emergent light of the zeroth orderis unobstructed while that of the first order experiences Braggreflection. Light of all negative diffraction orders is obstructed andlight of all diffraction orders exceeding one is greatly attenuated as aresult of inefficient reflection (since the Bragg condition is notsatisfied) and as a result of obstruction. Coefficient R depends uponthe material comprising the grating layer and is determinedexperimentally. R is primarily influenced by the Bragg reflectioncoefficient, which depends upon angle of incidence a.

A thick amplitude grating of this type can be made by exposing asilver-based, high contrast, high resolution photographic plate to twocoherent beams of light TABLE ll.-INTENSITY vs. DIFFRACTION ORDERRelative intensity r* in Grating diffraction order Ratio dominantRulings Blaze intensities, Calculated* per mm angle, 2 1 0 l 2 l /l,,6,,/(9 9,)

200 600 6 15 100 9] .91 .55 360 1022 5 I3 100 68 5 .68 .52 400 1354 5 ll52 100 4 1.92 .64 600 1727 5 I7 100 57 .57 .53 I200 2645 I00 26 .26 Al rFor angle of incidence a as prescribed by Eq. 1. *For wavelength 546lA.n L566.

Relative intensities of light output in each diffraction order wasmeasured with a photomultiplier type photometer, and given in TABLE II.The ratio of intensities of the two dominant diffractions, l ll wasfound to vary around the ideal value of unity. A graph of 1 /1,, vs. t9,/(0,-0,,) was plotted and the faired curve crossed I /l, l at p Onemight anticipate equal sharing of energy when the Snell refractiondirection bisects the angle between zeroth and first order diffractions.Actually, equal sharing is achieved when the refraction is slightlylarger than this, namely, 0.57 of the angle. In selecting or designing areference grating, this should be a primary criterion. When selectingfrom listings of standard gratings, blaze angle should be chosen suchthat Eq. l0

is most nearly satisfied. If the optical system allows variationofwavelength, A can be chosen to improve or perfect the fit to Eq. l0.Thick Amplitude Gratings Double-order dominance can be achieved with abarand-space grating if the thickness of the bars is sufficient to blockor grossly attenuate all diffraction orders except the zeroth and firstorders This condition is illustrated in cross-sectional view in FIG. 7,wherein 71 is the grating layer on transparent substrate 72. Opaque bars73 and transparent zones 74 comprise the grating.

intersecting at an angle 20:. For such conditions, when or is near 30, Ris approximately 0.9. Other photosensitive materials can also be used.Pitch g of the index grating thus produced is given by.

Thin Amplitude Gratings If the thickness h of opaque bars is smallcompared the only diffraction orders contained in the transmitted lightare the zeroth and first vordersJThe result is double-order dominance.For practical applications, the

intensities of these dominant orders are sufficiently close to beingequal when the width of opaque bars equals or exceeds the width oftransparent spaces.

Thin amplitude gratings that satisfy these requirements may be producedby photoetching the array of bars in an'evaporated metal coating on aglass substrate. The photo-sensitive resist may be exposed to.

two beams of coherentlight intersecting at an angle 2a.

to generate the grating pattern. 5

Reflection Configuration A reflection configuration equivalent to thetransmission configuration of FIG. 5 is shown in FIG. 8, Incident light81, at angle a, is divided by index grating 21 into two dominantcomponents of orders and 1. After reflection at scale grating 13 andsubsequent transmission of the index grating in the zeroth diffractionorder, dominant components 82 and 83 emerge and combine to form themoire multiplication pattern. As in the case of transmission, thesecomponents form group s 0. Rays that experience diffraction orders (0,3/2 s, 0) and l, 3/2 s, 0) at the three successive encounters with thegratings, constitute the adjacent s-groups. As in the case oftransmission, observations shound be made in the s-group that exhibitsoptimum fringe contrast.

PREFERRED IMPLEMENTATION 1. In a preferred implementation for moirefringe multiplication by a factor of 50, the optical elements arearranged as in FIG. 2. Scale 13 is a precision Ronchi ruling of 40rulings per mm (or 1,000 per inch), formed as evaporated chromium barson a glass substrate. The ruling is 20 mm (or 0.8 in.) long by l m(or 40in.) wide. Index grating 21 has 2,000 rulings per mm (or 50,000 perinch), and has an active area 15 mm (or 0.6 in.) square, bordered by anopaque mask. It is an evaporated chromium bar-and-space grating on aglass substrate, constituting a thin index grating. The gratings areilluminated by a collimated beam of blue light centered at 4400A, of 25mm. (or 1 in.) diameter,incident at a 26.l. The receiving systemisolates and processes transmitted light of group s 0. The receivingsystem, conditioning and counting system is arranged such that the leastcount corresponds to onefourth of a fringe. The system discriminatesbetween forward and backward motion. Sensitivity is one-half micron (or20 microinches) per fringe, and one-eighth micron (or microinches) percount.

2. In another preferred implementation, all conditions are the same asprescribed in Case (1) above, except the index grating is a phasegrating with blaze angle of 43, formed in a transparent material ofrefracfive index he W391i? ssqsit vit i de t a 3. In another preferredimplementation for multiplication by 50, the elements are arranged as inFIG. 4. The receiving system isolates and processes reflected light ofgroup s 0. The gratings are identical to those prescribed in Case (1)above, and the resultant sensitivity is identical. 0

4. In another preferred implementation, all conditions are the same asprescribed in Case (3) above, except the scale length is 25 cm (or in.)and the receiving, conditioning and counting system is arranged suchthat the least count corresponds to 1/20 fringe. Sensitivity is one-halfmicron (or microinches) per fringe and 0.025 microns (or I microinch)per count.

5. In another preferred implementation for multiplication by lO,conditions are the same as prescribed in Case (2) above, except for theindex grating and angle of incidence. Here, the index grating has 400rulings per mm (or 10,000 per inch) and a blaze angle of 95. Angle ofincidence is 5 Sensitivity is 2.5 microns (or 0.0001 in.) per fringe,and 0.6 microns (or microinches) per count.

In prior art attempts are made to enhance the resolution of measuringapparatus by treating electrical output signals in a special way. Incontrast to this, the present invention enhances resolution of measuringapparatus by treating the optical signal itself. More specifically,signal generating systems utilizing gratings (here called moire systems)are governed by the relationship I a cos 21-rBa /g or else, by anothercyclic function of the same frequency, where I is intensity of the lightoutput signal a is a constant B is an integer called fringemultiplication factor g is pitch of the scale grating d is displacementof scale grating relative to the index grating.

In traditional designs, B l, or in special circumstances it may equal asmall integer, for example [5 2. With fringe multiplication, however, [3may be a relatively large integer; B 50 was specified in severalpreferred implementations of the method. Accordingly, for a given scaledisplacement d, many more cycles of light oscillation are produced whenfringe multiplication is utilized. Then, high resolution of displacementmeasurements can be obtained even with relatively simple signalprocessing techniques.

It will be obvious to those skilled in the art that various changes maybe made without departing from the scope of the invention and theinvention is not to be considered limited to what is shown in thedrawings and described in the specification.

What is claimed is:

1. Apparatus in which the position of a table is determined by moirefringe multiplication comprising a table and base device,

a coarse scale grating attached to one element of said device comprisedof said table and its base,

an index member in the form of a diffraction grating of frequencyapproximately a whole number B, exceeding three, times the frequency ofsaid scale grating fixed to the other element of said table and basedevice in juxtaposition to said scale grating,

an illumination means radiating a systematic bundle of rays obliquelyupon said index member,

whereupon said index member causes two dominant components of said raysto emerge with approximately symmetrical obliquities in the spacebetween said index member and said scale grating,

said two dominant components being subsequently diffracted by said scalegrating, causing them to emerge from said arrangement of scale gratingand index member as two sets of subcomponent rays, with a multiplicityof pairs of said subcomponent rays, each pair comprised of a member fromeach set, traveling in a systematic sequence of diffracted directions,

optical filtering means which subsequently intercepts said pairs ofsubcomponent rays and allows only one pair to proceed, namely, a favoredpair that proceeds in a direction approximately normal to the plane ofthe scale grating,

observing means that intercepts the said favored pair and senses thestate of interference generated by said favored pair, said state ofinterference being manifest as fringes called moire fringes,

and interpretation means to compare and indicate relative position ofsaid table in relation to said base from changes of the said moirefringes.

pa t-t hibits dominant intensities in transmission for two neighboringdiffraction orders.

6. The apparatus of claim 4 in which said index member is an amplitudegrating that exhibits dominant intensities in transmission for twoneighboring diffraction orders.

7. The apparatus of claim 4 in which said index member is a phasegrating that exhibits dominant intensities in transmission for twoneighboring diffraction orders.

8. The apparatus of claim 1 in which said scale grating is abar-and-space type phase grating.

-9. Theapparatus of claim 8 in which said index member is anon-symmetrical blazed phase grating that exhibits dominant intensitiesin transmission for two neighboring diffraction orders.

10. The apparatus of claim 8 in which said index member is an amplitudegrating that exhibits dominant intensities in transmission for twoneighboring diffraction orders.

11. The apparatus of claim 1 in which said scale grating is asymmetrically diffracting phase grating.

12. The apparatus of claim 11 in which said index 2. The apparatus ofclaim 1 in which said index member is a non-symmetrical blazed phasegrating that exhibits dominant intensities in transmission for twoneighboring diffraction orders.

3. The apparatus of claim 1 in which said index member is an amplitudegrating that exhibits dominant intensities in transmission for twoneighboring diffraction orders;

4. The apparatus of claim 1 in which said scale grating is abar-and-space type amplitude grating.

5. The apparatus of claim 4 in which said index member is anon-symmetrical blazed phase grating that exmember is a non-symmetricalblazed phase grating that exhibits dominant intensities in transmissionfor two neighboring diffraction orders.

13. The apparatus of claim 11 in which said index member is an amplitudegrating that exhibits dominant intensities in transmission for twoneighboring diffraction orders.

14. The apparatus of claim 1 in which said index member is a phasegrating that exhibits dominant intensities in transmission fortwoneighboring diffraction orders.

15. The apparatus of claim 1 in which said favored pair of subcomponentrays emerges by transmission through said scale grating after one memberof said favored pair experiences diffraction orders a, ,8/2 5 while theother member experiences diffraction orders a l, -,B/2 s, where the twoterms separated by a comma represent diffraction orders at the saidindex member and said scale grating, respectively, and where a is asmall whole number including zero and s is a whole number between B andB.

16. The apparatus of claim 1 in which said favored pair of subcomponentrays. emerges by reflection from said scale grating and subsequentransmission through said index member and one member of said favoredpair experiences diffraction orders a, 8/2 s, 0, while the other memberexperiences diffraction orders a l, B/2 s, 0, where the three termsseparated by commas represent diffraction orders experienced upon encountering the index member, scale grating and again the index member,respectively, and where a isa small whole number including zero and s.is a whole number between -,B and B.

1. Apparatus in which the position of a table is determined by moirefringe multiplication comprising a table and base device, a coarse scalegrating attached to one element of said device comprised of said tableand its base, an index member in the form of a diffraction grating offrequency approximately a whole number Beta , exceeding three, times thefrequency of said scale grating fixed to the other element of said tableand base device in juxtaposition to said scale grating, an illuminationmeans radiating a systematic bundle of rays obliquely upon said indexmember, whereupon said index member causes two dominant components ofsaid rays to emerge with approximately symmetrical obliquities in thespace between said index member and said scale grating, said twodominant components being subsequently diffracted by said scale grating,causing them to emerge from said arrangement of scale grating and indexmember as two sets of subcomponent rays, with a multiplicity of pairs ofsaid subcompoNent rays, each pair comprised of a member from each set,traveling in a systematic sequence of diffracted directions, opticalfiltering means which subsequently intercepts said pairs of subcomponentrays and allows only one pair to proceed, namely, a favored pair thatproceeds in a direction approximately normal to the plane of the scalegrating, observing means that intercepts the said favored pair andsenses the state of interference generated by said favored pair, saidstate of interference being manifest as fringes called moire fringes,and interpretation means to compare and indicate relative position ofsaid table in relation to said base from changes of the said moirefringes.
 2. The apparatus of claim 1 in which said index member is anon-symmetrical blazed phase grating that exhibits dominant intensitiesin transmission for two neighboring diffraction orders.
 3. The apparatusof claim 1 in which said index member is an amplitude grating thatexhibits dominant intensities in transmission for two neighboringdiffraction orders.
 4. The apparatus of claim 1 in which said scalegrating is a bar-and-space type amplitude grating.
 5. The apparatus ofclaim 4 in which said index member is a non-symmetrical blazed phasegrating that exhibits dominant intensities in transmission for twoneighboring diffraction orders.
 6. The apparatus of claim 4 in whichsaid index member is an amplitude grating that exhibits dominantintensities in transmission for two neighboring diffraction orders. 7.The apparatus of claim 4 in which said index member is a phase gratingthat exhibits dominant intensities in transmission for two neighboringdiffraction orders.
 8. The apparatus of claim 1 in which said scalegrating is a bar-and-space type phase grating.
 9. The apparatus of claim8 in which said index member is a non-symmetrical blazed phase gratingthat exhibits dominant intensities in transmission for two neighboringdiffraction orders.
 10. The apparatus of claim 8 in which said indexmember is an amplitude grating that exhibits dominant intensities intransmission for two neighboring diffraction orders.
 11. The apparatusof claim 1 in which said scale grating is a symmetrically diffractingphase grating.
 12. The apparatus of claim 11 in which said index memberis a non-symmetrical blazed phase grating that exhibits dominantintensities in transmission for two neighboring diffraction orders. 13.The apparatus of claim 11 in which said index member is an amplitudegrating that exhibits dominant intensities in transmission for twoneighboring diffraction orders.
 14. The apparatus of claim 1 in whichsaid index member is a phase grating that exhibits dominant intensitiesin transmission for two neighboring diffraction orders.
 15. Theapparatus of claim 1 in which said favored pair of subcomponent raysemerges by transmission through said scale grating after one member ofsaid favored pair experiences diffraction orders a, Beta /2 + s whilethe other member experiences diffraction orders a + 1, - Beta /2 + s,where the two terms separated by a comma represent diffraction orders atthe said index member and said scale grating, respectively, and where ais a small whole number including zero and s is a whole number between -Beta and Beta .
 16. The apparatus of claim 1 in which said favored pairof subcomponent rays emerges by reflection from said scale grating andsubsequen transmission through said index member and one member of saidfavored pair experiences diffraction orders a, Beta /2 + s, 0, while theother member experiences diffraction orders a + 1, - Beta /2 + s, 0,where the three terms separated by commas represent diffraction ordersexperienced upon encountering the index member, scale grating and againthe index member, respectively, and where a is a small whole numberincluding zero and s is a whole number bEtween - Beta and Beta .